Help on writing symbols
symbol write
∧ /\
∨ \/
¬ !
F FF
T TT
⇒ ==>
⇐ <==
⇔ <=>
→ -->
↔ <->
∀ AA
∃ EE
≤ <=
≥ >=
≠ !=
+ +
− -
xy x y
x+1
y
(x+1)/y

And, or, not, and parentheses

We first use the following operators:

symbol read write as alternative
∧ and /\ and
∨ or \/ or
¬ not ! not

You may also literally write “and”, or even copy and paste the symbol “∧”. Of course, the same applies to the other symbols.

Please notice that “first ∨ second” really means “first or second or both”!

Let B mean that Bern is a nice city, G that Geneva is a nice city, and Z that Zürich is a nice city. Express the following claims.

Zürich is nice.

Zürich is bad. (Sorry for this, but also negative examples are needed ☹️)

Both Zürich and Bern are nice.

How about “Both Bern and Zürich are nice”? Does it mean the same as or different from “Both Zürich and Bern are nice”? Try it on the above and see whether it is correct! helpYou should try “B ∧ Z”.

Zürich or Geneva is nice (or both are).

All these three cities are nice.

At least one of these three cities is nice.

Zürich is nice but Bern is not.

Parentheses “(” and “)” may be used to express long claims. Here you can try how expressions are structured with and without parentheses. Please experiment with the example written below! Try it, then remove some parentheses, try again, add parentheses somewhere else, and so on. Currently only FF, TT and UU can be used here; sorry for that.

A parse tree Now it is your task to get the parentheses to the right place. Please add parentheses so that you get the same picture.

We now continue expressing claims.

It is not the case that both Bern and Geneva are nice.

Bern or Geneva is bad.

Please think about “It is not the case that both Bern and Geneva are nice” and “Bern or Geneva is bad”. Do they mean the same? Try the answer of one in the answer box of the other!

Either Zürich or Geneva is nice, but not both.

Often a claim can be expressed in a simpler form. For instance,

claim simplified claim
Z ∨ Z Z
G ∨ (¬G ∧ B) G ∨ B
B ∧ ¬B F
In mathematics, a claim may be undefined. For instance,
x
0
= 1 is undefined. It is important in programming and computer science to deal appropriately with undefined claims, and the feedback tool can largely do so. Unfortunately, the possibility of undefined claims is usually ignored in logic. Therefore, to keep things simple, today we mostly ignore the possibility of undefined claims.

In the last example, F means that the claim never holds. Indeed, Bern cannot be nice and not nice simultaneously. A claim that always holds is called a tautology and can be simplified to T.

symbol read write as alternative
F false FF false
T true TT true

Be sure not to type F and T as F and T! The latter mean, for instance, that Frankfurt or Torino is nice.

We use “⇔” to denote “simultaneously true”. So the above examples can be re-written as follows:

Z ∨ Z ⇔ Z
G ∨ (¬G ∧ B)   ⇔   G ∨ B
B ∧ ¬B ⇔ F

Simplify the following expressions.

¬¬Z  ⇔ 

G ∧ B ∧ G  ⇔ 

¬G ∨ G  ⇔ 

¬F  ⇔ 

¬(G ∧ B) ∧ G  ⇔ 

(B ∨ Z) ∧ (¬B ∨ Z)  ⇔ 

Here you may compare any two propositional formulae, whether they are simultaneously true! You may use any letters, like L for Lausanne.
 ⇔ 
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